One of the core principles of Kemalism is to be rather careful with foreign politics and avoid getting too much involved in foreign conflicts. It is an ideology which advocates avoiding risks here.
Joining the Axis is a highly risky move. If they win the war, you could gain a lot (if they are faithful and still want to fullfill their promises), if they loose the war, you loose the war with them.
Similarly joining the Allies too early in WW2 is highly risky, since it would mean war at your borders with highly superior opponents.
Maybe minor border corrections for minor help for one or another side in World War 2 is possible, but you need a government following a different ideology if you want Turkey joining World War 2 earlier.
If not direct involvement how about something like the Spanish Blue Division? A volunteer force raised for service with one of the sides.